A New Approach to the BHEP Tests for Multivariate Normality

Let X$_1$, …, X$_n$ be i.i.d. random d-vectors, d⩾1, with sample mean X and sample covariance matrix S. For testing the hypothesis H$_d$ that the law of X$_1$ is some nondegenerate normal distribution, there is a whole class of practicable affine invariant and universally consistent tests. These procedures are based on weighted integrals of the squared modulus of the difference between the empirical characteristic function of the scaled residuals Y$_j$=S$^{−1/2}$(X$_j$−X) and its almost sure pointwise limit exp(−‖t‖$^2$/2) under H$_d$. The test statistics have an alternative interpretation in terms of L$^2$-distances between a nonparametric kernel density estimator and the parametric density estimator underHd, applied to Y$_1$, …, Y$_n$. By working in the Fréchet space of continuous functions on R$^d$, we obtain a new representation of the limiting null distributions of the test statistics and show that the tests have asymptotic power against sequences of contiguous alternatives converging to H$_d$ at the rate n$^{−1/2}$, independent of d.